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We use renormalization group (RG) techniques to prove the nonlinear asymptotic stability for the semistrong regime of two-pulse interactions in a regularized Gierer–Meinhardt system. In the semistrong limit the localized activator pulses interact strongly through the slowly varying inhibitor. The interaction is not tail-tail as in the weak interaction… (More)

We use a multi-scale analysis to derive a sharp interface limit for the dynamics of bilayer structures of the functionalized Cahn–Hilliard equation. In contrast to analysis based on single-layer interfaces, we show that the Stefan and Mullins–Sekerka problems derived for the evolution of single-layer interfaces for the Cahn–Hilliard equation are trivial in… (More)

We employ global quasi-steady manifolds to rigorously reduce innnite dimensional dynamical systems to nite dimensional ows. The manifolds we construct are not invariant, but through a renormalization group method we capture the long-time evolution of the full system as a ow on the manifold up to a small residual. For the parametric nonlinear Schrr odinger… (More)

A wide class of problems in the study of the spectral and or-bital stability of dispersive waves in Hamiltonian systems can be reduced to understanding the so-called " energy spectrum " , that is, the spectrum of the second variation of the Hamiltonian evaluated at the wave shape, which is constrained to act on a closed subspace of the underlying Hilbert… (More)

We show the soliton solutions of the integrable Manakov equations exhibit an instability under arbitrarily small Hamiltonian perturbations. The instability arises from eigenvalues embedded in the essential spectrum of the associated linearized operators; these eigenvalues are dislodged by smooth perturbations. Speciically we consider perturbations which… (More)